Possible post-Brexit border scenarios
Faced with the desire to maintain the "status quo" in the movement of goods between Northern Ireland and the Republic after Brexit, there are clearly two possible future scenarios, namely: to ask the EU to allow the Republic to apply UK-type controls on non-British/Irish citizens entering the Republic; or for mainland Britain to vet travellers coming from Northern Ireland in the same way as those coming from mainland Europe.
Considering the likely resistance from the EU to proposal one and the importance to both north and south to the maintenance of an "open border", it would appear, in theory at least, that rigorous checks at British ports of non-British/non-Irish travellers coming from Northern Ireland and the Republic would be the most sensible and acceptable way forward.
Post-Brexit this latter proposal might cause a temporary influx of EU and non-EU immigrants into mainland Ireland, but the message of their inability to enter mainland Britain via Ireland would soon sink in and bring about a closure to this "back-door" route into Britain.
Of course, a second outcome might well be that the Republic adopt UK-type controls themselves.